 Okay, welcome to your 10th tutorial session and last week it was on Wednesday, sorry, it was our session number 9 and we were covering binomial distribution and Poisson distribution. So today we're going to continue and do the activities relating to the binomial distribution and the Poisson distribution and also we're going to look at the counting rules but I'm not going to draw too much on the counting rules. We're just going to do two exercises and I'm just going to show you on the calculator please make sure that you are muted all the time. So yeah, so let's begin with I remember we talked we spoke about the counting rule and we said the the first counting rule which is the power rule it says if anyone of the k different mutually exclusive and collectively exhaustive events cannot care on each of the n trials the number of possible outcome will be equals to k to the power of n which means it's the number of outcomes to the power of the event. So we said if we have a die and we roll that die three times then the possible outcome that can happen will be 260 and we also looked at the counting rule number two which is the multiplication rule which says if there is if there are k nth events which is k one event on the first trial k two events on the second trial and k nth events on the fourth trials of and so forth and so forth up until the ninth k the ninth trial event and if we want to know how many possible outcomes we can have out of choosing any of the scenarios and we looked at an example of if you have to go to a park a restaurant or a movie and you have three parks for restaurants and six movies how many possible outcomes or how many possible ways can you go and visit all of the three environments and we said it will be three times four times six which will give you 72 different ways I am going to apologize for this because I am visiting in Pretoria and I'm staying in the location you know locations there is music there will be music playing but I hope it will not disturb our our recording as well okay the fact counting rule is the factorial rule which tells us the number of ways we can arrange items and we looked at how many number of ways can we arrange the five books on the bookshelf and we said it can be n times if there were five books then it will be n times four times three times two times one and it will give us 120 possible ways that we can arrange those five books then we also looked at counting rule number four which is permutation and with permutation we said there is an order or preference in terms of how you do things therefore with permutation tells you the number of ways you can arrange objects from n objects in an order or when you have a preference and we used an example of having five books that are we going to put on three shelves and we wanted to know how many number of ways can we place those books in an order manner and because we we we have an order in terms of how we're going to do things therefore we use permutation and permutation in n px which is n factorials divided by n divide n factorial divided by n minus x factorial and I said when we come to this part actually I wanted to show you how to do calculate the factorials for those who don't know or who are lost when we were doing them the question so I can actually go back to the first one which was the this one the factorial so factorial on your calculator this is a case your calculator those who are using a case your calculator your factorial is on top of your x to the power of a negative one it's an x factorial it's not an n it's an x factorial to calculate five factorial you just press five you press shift you press the button where x factorial is at and you will get five factorial and then press equal and that will give you your factorial and those who are using a case your you do those steps those who are using a sharp calculator with a sharp calculator sorry about that we need to bring back the slide with a sharp calculator we also want to calculate the factorial so your sharp calculator those who have it that looks like that that has the green letters and and orange letters on top of the buttons your n factorial you need to look for it you need to look carefully it is orange and it's on button number four and it is an n factorial also you go and say five and go second function because it's written in orange and then you press four and you will have n factorial s five factorial and then you press equal and that will give you hundred and twenty and that's how you're going to do your factorials to do permutation so when we answer question on permutation in this instance we know that we are arranging five books in an order uh in and put them on the three uh bookshelves so we you know how to do the factorial so i'm not going to do that part i'm going to show you to how you can do the npx on the case your calculator your calculator and then we first going to put five because five is our n so we say five and then we need to press the npx button it is on if i can look for it it's on multiplication button so then it's written in orange i'm going to press shift and then press multiplication and then i need to press three and i have npx or r and i press equal and that will give me 60 different ways similar on a sharp calculator to get the same answer here our calculator uh five is our n and our npr is on button number six so we go into press five first and then press second function press button number six and then press three and press equal and that will give you 60 and that's how you will calculate your counting rule the other counting rule that we learned was the combination with combination order that's not a meta or there is no preference in terms of how you do things so combination is the number of ways of selecting x object from an n object irrespective of order or preference and we use combination and we use this example where we said when we have five books and and going to read three of them the same question as we used the last time it's five and three we have five books and we can place them or read them uh three three books that we're going to select three books that we're going to to read so because yeah there is no order in terms of how we're going to read those books or there's no preference in terms of how we're going to read those books then we use combination and we found that that combination is equals to 10 on your cashier calculator i'm going to go clear cashier calculators your n n cr is on the button division and you do the same five is our n shift division and three equal and that is five combination of three and the answer is ten similar you will have on the sharp calculators those who are using sharp calculators you're going to press five or your n cr is on button number five but here we need to press five for n second function and press the button five again and press button three and say equal and there we get ten and that is what i wanted to show you in terms of using your calculators okay then we went on and and looked at the binomial distribution and we said with binomial distribution the events are sequential uh trials that happens with n with n parameter and there are always two outcomes possible outcome for each trial and we'll say one outcome is a probability as a success and one outcome is a failure and from getting a success and a failure we can calculate the probability of success and the probability of failure and we know that the probability of success is denoted denoted by the pi and probability of failure will be one minus pi which is one minus the probability of success and both of the events or both of the trials needs to be mutually exclusive and they also have to be collectively exhaustive and each event needs to be an independent event that happens the other thing i need to mention as well i think it's not part of the slides is that a binomial distribution comes from remember it's part of the discrete distribution and a discrete distribution is a discrete random distribution random it comes from a discrete random variable therefore it uses either the measures that can be or it uses measures that can be counted and if it's measures that can be counted those measures are an integer measures as well i'm gonna look at the exercise that we did then we also looked at the characteristics of a binomial distribution in terms of the mean the variance and the standard deviation and we said the mean of a binomial distribution is your expected value which is your n times the probability of success your variance which is the population parameter the variance it's given by n times the probability of success times the probability of failure and your standard deviation is the square root of your variance and you need to know how to calculate all of this then we went and looked at the probabilities how we find the probability and we said with a binomial distribution we use the formula and you need to know that we can use this formula as well to answer the questions on the binomial distribution the challenge becomes when you have a greater than or equal or less than or equal questions where you have to calculate for multiple probabilities at a point then this formula becomes complex because it will take you forever and you don't have enough time in the exam to calculate the probability of x is equals to zero plus the probability of x is equals to one plus the probability of x is equals to three or two probability of x is equals to three probability of x is equals to four and so forth it's gonna take you forever so in state of us using the formula we're going to use the table and we said on the table the tables are designed in a way that the top probabilities that you get you're going to use the n and the x value on your left and the bottom probabilities the bottom probabilities at the at the bottom here there are probabilities remember I said because this table is on one page this one is on another page the probabilities are cut off at the end of the other page because they're meant to be one page which has all these probabilities so the probabilities at the bottom of the page correspond so all these probabilities at the bottom correspond to all the n and the x value from this side so it means when you read the probabilities of more than 0.99 or more than 0.55 you will use the bottom part to read your table if you're using probabilities of 0.1 between 0.01 and 0.5 oh yes 0.45 I'm going to say 0.45 you use this so let's say from here to there you use the left from 0.9 to 0.45 you use the right 0.50 you can use either one it will give you the same answer because it's 50-50 and we know that the sum of all probabilities should be equals to zero even within the whole the sum of all these probabilities should give you one so that if you add all the probabilities within the let's say it is in this n the sum of these probabilities in the n should give you one the sum of this probability of success and the sum of probability of success at the top should give you one so we need to always remember that okay so that is the table that we're going to use for the binomial so when we do the exercises later on then we need to know how to use the table then we looked at Poisson distribution and we said that Poisson also because it's from a random discrete variable as well but with Poisson it uses the area of an opportunity unlike with binomial where it uses the probability of success here we use the area of opportunity which is our lambda and we know that in terms of the characteristics of a Poisson as well the the expected mean or the lambda or the average is the same as the variance remember that the mean the expected value or expected events the average the lambda the variance are the same and the standard deviation is the square root of your variance or is the square root of your lambda and we did the exercise and we said also with Poisson you need to know how to also use the probability formula to calculate the probability of a Poisson and also like I mentioned with the binomial that it becomes very complex when you have multiple probabilities that you are calculating then we rely on the table but sometimes in the question in the exam they might ask you the Poisson formula without you calculating it to the extent but just to make sure that they understand that you know what Poisson formula looks like and how do we calculate the Poisson probability as well so you must remember the formula that is the Poisson formula and we can use it to calculate the probability of an event or we can use a table and we also looked at the table and we said all the tables are broken down by the lambdas and all the lambdas have different x values from zero up until the end they differ within the table so one table might have lambda the lambdas and then have the x of zero up until seven the other one might have zero up until 12 the other one might have zero up until 14 and so forth so you need to be very careful when you work with the Poisson distribution you must go to the exact lambda table in order for you to know the value of your x your actual value of your x okay not going to do that exercise we oh sorry I don't want to close because I've done all the activities in here so today's session we need to start with today's session sorry about that I need to open the session today's session should be part two activities apologies for that okay so let's start exercise number one suppose there are three positions to be selected out of 12 for a committee how many possible arrangements can they be that is your exercise what is this question what the question is asking us is this this accounting rule question and they are telling us that there are three positions that needs to be selected out of 12 is this a combination or permutation was there order or preference talk to me or you can post your answer on the chat remember we we're still using the chat as well or unmute and talk to me I think we are not I think the silence is we are confused on what we should be doing but I've asked you a question is this question I said I gave you the hint I said this is accounting rule question and then I asked is this a permutation question or a combination question and then you were you were quiet then I went on and I asked you if you read the statement is there order or preference given on that question that's the question I'm asking is there order or preference there is no order so if there is no order then is this a permutation question or is this a combination question this will be a combination question it is a combination question because combination says there is no order or preference in the way you arrange things or the way you do things so this will be your n c r question so based on the question what is your n is your n not 12 your n is 12 what is your x or r that would be and calculate your combination I get 220 is that correct are we all in agreement with 220 yes I'm going to only use the cash your calculator here so let's see so we set this is n c r which is five so on our calculator we go and we say no why are you not correcting me now when I'm typing the five and the three is it should be 12 and three so we first press 12 then we go shift and we go division and we press three and equal and the answer will be 220 number two there is a president the secretary at terrazara to be selected out of a 12 committee how many possible arrangements can they be is there order or preference given no yes how many of you are saying yes how many of you let let's start with the the yeses I had one yes are there other people who agree with the yes yes yes are there other people are there other people who are agreeing with the no those who are saying yes there is preference why are you saying there is preference isn't that they are ordered from precedent to treasure as asking isn't that there's order because it's from precedent all the way down to treasure another reason anyone or are you all agreeing with what she just said yes I agree okay yes I because there is an order in which they listed them as well but also they gave a preference in terms of which positions they need to be selected they could have just said let's select three people the same way as we had in the first one they said suppose there are three positions to be selected out of this there is no order or preference in terms of who they are selecting they just wanted to select three people here they say there is a president that needs to be selected a secretary that needs to be selected a treasurer that needs to be selected they give you a preference in terms of what positions and also they give you the order in which you will have to select those or the order in which the committee will be constituted and that makes it if there is an order this question then it is a permutation room are we calculating combination or permutation here we calculate in permutation so it will be n pr which is almost the same as what we had before our n is 12 and the three positions are our x so n pr or np3 and calculate what is the answer the answer is 1,320 1,320 so we say 12 shift n pr on a multidimcation 3 and equal and we get the answer of 1,320 okay and that's how you will answer the questions in terms of the permutation combination also if they give you in terms of the other multiplication rule ask your ask yourself questions is the question all the statements given did they tell me that there is an order or is there clearly can I see that there is an order of how things happen or are they no order or preference in terms of how things need to happen or so you need to ask yourself all those questions or are they giving me multiple things that I need to arrange or do then it's multiplication rule or are they giving me only one thing and they telling me how many possible ways I can get to that then it is your factorial or are they telling me this to the power or I need to do x amount from the same I have repeat the same events as often as possible then it is your power because if if I toss a coin three times I am creating I'm using the same coin which has two outcomes which is the head or the tail but I'm creating three multiple events out of it so then it means I'm twisting it three times out of those two outcomes it will be two to the power of three so you just need to ask yourself all those questions as well okay and that is the counting rule so now let's move on to the binomial now in the exam let me just say it this way in the exam the way they will ask you questions they follow the order in which we do them in or in order they are in the study guide or in your chapters so when you deal with questions from discrete probability after the discrete probability you will be dealing with questions about binomial but you need to be able to identify the question because you are not going to be told sometimes you will not be told that this is a binomial question you need to understand that you need to know that now I'm talking about the binomial and I'm saying this because when you answer the question sometimes you will answer the basic probability questions where we also use percentages and calculating probabilities and proportions then immediately after that you find questions about discrete probability and you will know that with the discrete probability you will get the table with your x observations and your the corresponding probability after that you will get the questions about the binomial distribution and those questions about the binomial it can be calculating the binomial distribution or unpacking and understanding the concept and characteristics of a binomial so it can be in both ways it can be calculated the probability calculate the variance the standard deviation or the mean or they can ask you about the content remember the past week someone asked instead are they asking you how many questions in in your exam paper will be calculation and how many can be a theory is a mixture so you need to be prepared to know both you need to know the content the content and you need to know how to calculate as well so one of the question is this which one of the following statement is incorrect one if the value of a variable depends on an outcome of an experiment the variable is called a random variable is that true or false is it correct or incorrect is number one correct talk to me are you all muted am I not audible enough Lizzie isn't it false I will also it's false because it says random variable if you are conducting an experiment remember now you are in the binomial in the discrete environment ir regardless of that as well so when you create team an experiment you always get random events happening isn't it and those random events have characteristics and those characteristics are variables isn't it so number one is correct because number one says if the value of a variable depends on an outcome so we know that that value depends on an outcome of an experiment that is happening the variable will be a random variable and that is correct because it will be random when you toss a coin it will randomly drop on a head or a a tail you're not going to just place it on a head and say but I've tossed it you tossing it and it will choose the randomly on which side that coin will land it will choose by itself random so when they talk about random you know that the the events are also in a way independent because there is no influence from another another thing also so it's a random event that is happening okay so number one is correct I'm going to skip number two I'm gonna go down to the bottom one so that we we we talk about the things that we already learned and know about so number five says the probability of an event is always on the range zero to one that is the probability of an event lies between zero and one is that correct yes that's correct that is correct I'm going to skip four I'll go to three the number of outcomes obtained when two dies are rolled is equal to 36 no it's not 36 72 when two dies are rolled a die has how many outcomes speaks and die has six outcomes yes and if there are two of them it's 36 six to the power of two which means is six multiplied by six which is 36 that is correct I'm gonna go back to number four the mean of a random variable that follows a binomial distribution with the parameter n and pi probability of success is equal to n multiplied n times pi is that correct the mean or the expected value of a binomial what is the formula of a binomial distribution the mean of a binomial distribution it's n multiplied by pi is that correct this question is correct because that's all what they are asking you the mean of a random variable that follows a binomial distribution with the parameter n and pi is equals to n times n the mean is equals to n times pi or n times the probability of success that means that is correct number two a discrete random variable takes on any numerical value with a certain interval here we're talking about discrete random variable is that statement correct or incorrect just to help you because it's the only one that is incorrect in this instant a discrete random variable takes any value but not from an interval but of an integer because a discrete random variable comes from a counting process because this comes from a counting process and only those that comes from a a measured process which is like your your temperature your height they will take an interval value so the interval is those that comes from a continuous variable and we will deal with those ones in the next chapter chapter or study unit six when we do normal distribution continuous variables so the only one that is incorrect is number two okay your exercise three suppose that an admission test for a certain university is designed so that the probability of passing is 45 percent find the probability that among five candidates who take the test more than three will pass how do i know that this is not a basic probability question if you look at this they're asking you find the probability among five it will not be a basic probability it will be because they give you your n they give you your probability so they give you your n they give you the probability and they give you they're asking you to calculate the probability of pass and since here they say the probability of passing then you can assume that this is your probability of success and then they're talking about binomial distribution so this will be your probability of success that is your probability of success and your five is your n and the question asks find the probability that among those five candidates who took the test more than three will pass so it means what is the probability that x is greater than three find that probability i'm going to do this with you going to open the table so let's go to the table you need to find the table table e six there we go need to rotate it rotate okay going back to our question our question set five so i need to go find n five the probability of success of zero comma four five because it's 45 percent and i need to find the probability that x is greater than three that was the question asked find the probability more than three will pass so the first thing remember that we need to look at is the probability of success so the probability of success says 45 45 is at the top so it means my n i'm going to use the n on my left and because it says the probability of x great greater than three then it means we need to find the probability the probability of x is equals to four plus the probability of x is equals to five if i go to n i must be there the n will end with this our x values will end with the same as the probability the n value so the last value of the n will correspond to the last value of your x so it means i'm only calculating the probability of x plus four plus the probability of x equals to five so going to the five block is this probability of 45 then i'm only looking at the last two values so i'm just going to say zero comma one one two eight plus zero comma zero one eight five and we go to our calculator and add the point one one two eight plus point zero one eight five equals and the answer we get is zero comma one three one three going back to our question this is equals to zero comma one one two eight plus zero comma zero zero zero comma zero one eight five which is equals to zero comma one three one three which is option number three option number three and that's how you answer the questions any question are you happy yes thank you okay autism South Africa has found that 50 percent of the people with autism ASD struggle with social interaction assume we randomly select six people living with ASD probability of success consider the statement A to C A the expected number of people with ASD struggling with social interaction is three B the variance of the number of people with ASD struggling with social interaction is 1.5 C the standard deviation of the number of ASD struggling with social interaction is 1.25 which statement all statements are correct from the above so number one you need to go and calculate the expected mean which is pi times n or n times pi number two you need to calculate the variance which is n times pi times one minus pi bigger and number three you need to go calculate the standard deviation i'm just going to use that function standard deviation which is the square root of pi or n times pi times one minus pi minus pi so you have five minutes to do the calculations and then we recap if you want to ask a question i'm going to get a glass of water just now i'll be back are we winning are we winning yes done okay let's give those who are still busy chance are we done yes okay see four people answered okay let's do it together a so a one has to calculate the expected value which is n times pi what is our n six six and times our pi or probability of success which is 0.5 and when you calculate this what do you get the variance is pi times n one minus pi which is six times 0.5 times one minus 0.5 and what do you get 1.5 1.5 1.5 and we know that the standard deviation is the square root of a variance therefore it is the square root of 1.5 and what do you get 1.2 25 1.2 25 so which one of the statement is incorrect that is oh sorry it's correct that is incorrect that is incorrect that is incorrect therefore number five is the correct answer next question autism South Africa found that we've read the statement before we know that this is our probability of success which is 0.5 and this is our n which is equals to six what is the probability that only three people how do we represent that probability we need to find the probability of x x is equals to three and you can use the formula ncr pi to the power x one minus pi to the power n minus x or you can use your table so go ahead and use the table and I will go ahead and use the formula on here so okay so I'm going to assume that caraboy has confused the table that is why yes I used the table she was able to find the answer quickly and for some of us who are still busy with come on what did I do now with calculators it might take us long okay so I'll do it on the calculator I've already put in the building blocks so that we don't have to waste so much time on it so I have filled in my ncr remember the in front the ncr ncr is the same as n factorial divided by n minus x factorial of the binomial instead of me writing all this I can just represent it as the combination formula so the combination six shift combination three gives me 20 times 25 to the power of three equals give me zero comma one two five and it will be zero comma one two five I gain on this because one minus zero comma five it will be zero comma five to the power of three will still give me the same answer as the same as the previous one so 20 times zero comma one two five times zero comma one two five gives me zero comma three one two five that's what I get which is option three when we go to the table we are looking for delete this one first okay so we were looking for any six we need the probability that x is three and our pi is zero comma five so we go to our pi zero comma five we can either use this side or we can use that side it doesn't really matter because it's 50 because yeah at the bottom is zero comma five zero as well so we can go here and look for where x is three and we will still find the same answer and we go this side where x is three you will still find the same answer because there it is 50 50 so you will get the same answer so and that's how you will use the table it only works for both sides if you are using zero comma five zero otherwise if it's a bigger value than zero comma zero five we need to use the bottom and the right if it's less than we use the top yes you are on five and not six I'm on five not six yes you are right sorry my bad okay it's fine sorry six three I'm sorry my hand is not steady zero comma three one two five so also on five it gave me the same answer and also if I'm coming from six I get the same answer on the okay so moving on to question number six which of the following statement is incorrect poison and a binomial distributions are discrete probability distribution is this true or false that is true a poison distribution is characterized a characterized by one parameter namely the mean per interval or the average per interval or the distance of volume denoted by lambda is that correct true that is correct I'm going to skip three I'm going to go to four a binomial distribution or a binomial experiment the random variable x takes on integer values from zero up to n that is x of zero one two up until n where n is the number of sample size or the number of trials is that true or false true is true if you are in doubt in terms of the questions as well you can use your table to just look at the table because the table will guide you they start at zero to n value because it ends at the n is three it will end at n of three if n is six it will end at n x your x value will end at at six and because these are integers not interval values they do not have decimal values okay that makes it correct the standard deviation of a poison is equal to the square root of the expected value is that correct remember the expected value here is lambda since we said it there the mean or the average or the expected value or interval or distance so the standard deviation is equals to the square root of the expected is that correct that is the formula of the standard deviation is the square root of your mean so therefore it means this is correct number three a binomial distribution is characterized by one parameter namely the probability of success denoted by pi is that correct no it's not correct no it is not correct because it is characterized by two parameters which is n and pi the sample size and your pi this is incorrect so that is the answer that you were looking for now let's calculate the probabilities suppose that 10 percent of butterflies have damaged wings if a random sample of five butterflies is selected what is the probability that none of the butterflies have damaged wings you know that is the probability of success that is your n you can calculate what are we calculating x is equals to zero we're calculating x is equals to zero therefore you can also use the formula you can use the table if we use the table i'm going to go to the second one rotate we're looking for n pi of 10 percent which is zero comma one zero and the probability that x is equals to zero what is our n our n is five so if our n is five then we need to go back to the to the first one sorry we didn't have to come here my bad so our n is ten n is five i is zero comma one zero you need probability that x is equals to zero do we have an answer it should be the quickest one zero where n is five probability of zero comma one where x is zero zero point five nine zero five nine zero five that will be zero point nine five zero five that probability there that one that is the one that we're looking for that one zero point five nine zero five so zero point five nine zero five which is option number two using the binomial distribution if n is equals to five and the probability x is equals to three is zero comma one three two three the probability of success is what is pi that's what they are looking for hint on how to solve this you need to go to the table you need to go to where x is three in the table where n where n is five for that probability inside the table and go out so you will find that probability there where n n is five x zero one two three where x is three where that probability set you go out you will look for that probability at the top of the table it's okay three that's what you need to be looking for so let's go there so looking for n is equals to five x is equals to three and the probability that x is equals to three of zero comma one three two what is it one three two three so we got we come here we look for n is five x is three we go n with the table we just go on until we get zero comma one three two which is that what is that probability then we need the probability of success which is the answer is zero comma three zero so that is the answer we thought so if he's got the person or the tv mute themselves if the average number of adults with asd consulting with neuropsychologists per day is poison distributed with the mean of one point five so yeah we're talking about poison poison of one point five so it means we're going to use either the the formula to calculate the poison or we're going to use the table so what is the probability that on any given day a neuropsychologist will consult with only one adult what probability are we calculating yeah probability that x is it greater than or is it equal or is it less than x is equals to one x is equals to one because it says on any given day it's only one adult so we calculate in the probability that x is equals to one and you can use the formula which is lambda to the power x times e to the power minus lambda divided by x factorial or we can use a table or we can use a table so i'm going to use the calculations you go and find the value on the table and tell me if it's any of those i will complete the formula in the meantime no i'm not going to complete the formula in the meantime let's go find the probability on the table so we're looking for the poison we need to go to the table for poison which is broken down by lambdas so we need to rotate so our average it's equals to 1.5 so we need to go to a table that has 1.5 and there is our 1.5 what else are we given we are told on any given day it's we need the probability x is equals to one so we need to find the probability that x is equals to one so we just go to the table go here look for x is equals to one and that probability is equals to 0.3347 come back to our table this table value 0.3347 0.3347 and it is none of these values there so we know that that is not the question we're looking for need more information to calculate the probability not necessarily let's see one more option because we know we're given the probability we have still have the table to finish our lambda is 1.5 our x is one times our e is minus 1.5 divided by our one factorial is that what we're looking for that's what we're looking for and that is the answer that we're looking for any question absence of questions then we move on the average number of adults with ASD consulting with neuropsychologists we've read this sentence before is 1.5 what is the probability that at any given day a neuropsychologist will consult at least seven what are we calculating here we calculating the probability x is it less than or equal is it greater than or equal to seven therefore it means we need to find the probability we need to go fast to the table because we cannot determine how many of them we need to be calculating so we need to be calculating the probability of greater than or equals to seven so 1.5 we're back to the table we look for we need to calculate the probability that x is greater than or equals to seven so on 1.5 lambda average 1.5 greater than or equals to seven is here so we cannot do anything with those ones so greater than or equals to seven starts from there so we need to calculate the probability that x is equals to seven plus the probability that x is equals to eight plus the probability that x is equals to nine that's all what we need to be calculating which is zero comma zero zero zero eight plus zero comma zero zero zero one and plus zero i'm not gonna say zero comma zero zero zero zero which is zero comma zero zero nine that is the probability that we're looking for so which is option two probability that x is equals to seven plus the probability that x is equals to eight plus the probability that x is equals to nine if you wanted to use the the formula so you will say lambda x times e to the power negative lambda divided by x factorial plus and you do the same lambda x times e to the power minus lambda x and the last one will be lambda x i'll do we write lambda no lambda x e to the power minus lambda divided by x so you will substitute seven everywhere where you see x eight everywhere where you see x nine everywhere where you see nine and eventually in the end you should have zero comma zero zero eight and zero comma zero zero one plus zero comma zero zero zero zero and that should give you zero comma zero nine that is if you're going to use the table the formula next given the mean what is the variance of number of adults this should be easy should be straightforward what is the variance of a poison will be 1.5 which is the same remember that the mean which is the mu which is the expected mean which is the average is the same as the variance is the same as the variance is the same as the variance so that is the mean they are the same and they are the same as lambda they mean one and the same thing lambda next let x be a random variable representing the number of mistakes in a textbook suppose the mistakes okay at the average of two page and my average is the same as lambda is the same as the expected is the same as the mean that so average which is our lambda what is the probability that at most three in terms of a sign what are we trying to find here probability what is the probability pardon less or equals to three less or equals to three three therefore it means we will need to find because I know that it is two and it's three so it's fine we're going to need to find the probability that x is zero plus the probability that x is one plus the probability that x is two plus the probability that x is three so go into the table or you can use the formula so we need to go to two and two is also on the same page so three is somewhere there my hand is shaky where with me so you can also use your account so you can say always it starts with zero so you can say zero one two three and not do it the way I am doing so you need to add all of them because the lambda is two so we need to add zero comma one three five three plus zero comma two seven zero seven plus zero comma two seven zero seven plus zero comma one eight zero four sorry about that one eight zero four using my calculator point one three five three plus point two seven zero seven plus point two seven zero seven plus point one eight zero four equals zero comma eight five seven one there is your next question do you have the answer we're still using point two okay say it is option four so we say what are we calculating here the probability that x exactly five will just be equals to five zero zero one two three four five should be like be zero comma zero three six one which is option four and that is correct next assume that x is a Poisson random variable with the average of six which one of the following statement is incorrect so I'm going to go this way and go scroll to six so that we can have this table close by as well I'm gonna make it bigger and go to six six six is also right at the end at the edge so you will just do account zero one two three four okay so the first one we need to find the probability that x is equals to zero is it correct yes it's correct we're looking for the incorrect answer so that is correct because the probability that x is zero is zero comma zero two five number two the variance is two point four five is that correct you do not have to even think about it we are dealing with Poisson and it says the average is six the mean no it's not it is not correct but let's just double check the other probability that x is less than three it means you're going to add from two so you're going to do probability that x is zero plus the probability that x is one plus the probability that x is two because it says less than three and that will be correct because it's zero point zero four four six plus zero point one four nine plus zero point zero zero two five this list can I ask a question yes just to verify so the variance and the average would have been the same so the answer would have been six for option two for it to be correct yes okay thank you that's very fine so point zero zero two five plus point zero one four nine plus point zero four four six equals zero point six two so this would have been correct for greater than five so it means you need to start from here add all of them or you can say it is one minus the probability that x is less than or equals to five and therefore it means you just start from here add all of them up and subtract them from one of which I'm going to assume that that is correct if you do that because we already have our incorrect answer and yeah it says find the probability that x is between so for between you are going to see if I can remove the ink so for between and it does not include two and it does not include five therefore you're going to say x is equals to three plus the probability that x is equals to four which is just those two values but because it's not too many I can calculate quickly which is point zero eight nine two plus point one three three nine equals zero point two two three one which is correct in the exam you don't have to validate as soon as we get to the answer and you know that that is incorrect just keep that one if you're not sure just double check okay next question says suppose that the daily fake news follows a poison distributed with the mean of zero point two so it means we need to go to zero point two in order for us to answer all the questions let's go zero point two so we need to go to the table that has zero where you go so we can go to the site just minimize it zero point two and we can just so we know that our lambda is zero point two so we need to find all of this so the first one says the probability that there will be one fake news and we're looking for the incorrect one one probability are we looking for here x is equals to one x is equals to one and for a zero point two is that correct yes that will be correct number two the probability probability that there will be two fake news so what are we looking for here x is equals to two so where x is equals to two is that correct that is correct because where x is equals to two it's zero comma zero one six four number three the probability that at least three fake news what are we calculating here x is greater than or equal three therefore we need to add all of them the probability that x is equals to three plus the probability that x so if I look at this I don't have to worry about where they are zeroes because adding them won't make any difference x is equals to zero three x is equals to zero is equals to four but you could also just do x is equals to five is equals to six and x is equals to seven um so if you add zero and zero and zero you'll get zero but here we have zero comma zero zero one one zero comma zero zero zero so the answer will be is that correct yes it's correct yes it's correct and be correct okay the probability that there will be at least five fake news is equals to zero I'm sorry I removed option five on this one so there was a none none of the above I removed it because I didn't double check all these answers before I removed it there was number five every time you will get the number five the probability that there will be at least five fake news will be the probability that x is greater than or equals to five and that will be five plus six plus seven and we know that they are zero comma zero zero zero so that would have been correct the incorrect one will be none of the above that is the only instance where none of the above works in this instance because everything else is correct okay suppose that a number of the we already read this what is the probability that on any given day there will be no fake news what are we calculating here x is equal to zero x is equals to zero so if we use the table are we getting the answer where x is zero is zero comma eight one eight seven so it means that is incorrect that is incorrect and that is incorrect the only option left is for us to use the formula lambda x times e to the power negative lambda divided by x factorial and actually there was also here none of the above okay so coming here we need to replace the value of x and lambda our average is zero comma two to the power of zero because our x is zero times e to the power minus 0.2 divided by zero factorial so what is 0.2 to the power 0.2 to the power of zero is equals to one any number to the power of zero will be equals to one so that will be one times e to the power of e zero point two and what is factorial zero factorial is equals to one so our zero factorial is equals to one and therefore the answer one times e to the power of zero is the same as e to the power of negative so we can say this will be the same as as that e to the power of negative zero point two as you can see yeah it says is to the power of a positive I think there was an error on this question because then this the way it is at the moment it would have been incorrect but if there was a negative there it would have been the correct one otherwise then none of the above will also surface will also be the correct one in that instance because there is no negative there okay and you like with any other and that I give you I also give you multiple questions that you can go through and answer them on your own they we ended up on number 17 so you will need to go this is from here is your homework or your additional activities that you can do on your own for your preparation to do your assignment two and if you get stuck you can ask on whatsapp or on my unisa otherwise from 17 up until 25 why do I have 225 up to 27 you can do on your own I want to do 27 with this one minute that is left so 27 says I can just close everything and make the presentation 27 says a random variable x has a binomial distribution with unknown probability of success and the mean of two so they didn't give you the probability of success which is your pi but they give you the mean the mean we know what the mean is is the expected value they give us the expected value and we know how we calculate the expected value is n times pi and they give us the the the sample size so they say what is the probability of success if the sample size is equals to 8 so we know what our n is so because they gave us the mean you just substitute the mean which is true our mean is true our n is 8 and we need to find pi which is our probability of success and therefore our pi will be divide both sides by 8 8 will cancel out on the right and 8 will divide by this side so this divide by 8 divide by 8 8 and 8 will cancel and here will be left with the pi and here will be 2 divided by 8 which is the pi will be 1 over 4 and what is 1 over 4 is 0.25 the answer would have been option number 1 so i've done that for you so you have up until exercise 27 26 to do so it's 26 the 225s and 21 20 19 18 so if you do all those and you find any challenge let me know let's have a discussion on whatsapp or on my unisa i will let you know when the videos are uploaded with that thank you for coming through and if you have any question on the work that we did so far let me know otherwise have a lovely lovely weekend see you on wednesday when we do normal distribution also bring your statistic tables statistic out tables if there are no questions i have a question yes you may ask yes um with regards to the table the probability table with for binomial um if you're not uh given uh like for instance the probability that you have doesn't show on the table then what do you do do you just go ahead and just use the formula uh what do you mean the probability that they it doesn't show on the table yeah like if if you are given a probability of 0.62 for instance and there's no 0.62 on the table yes so if they give you this probability then you must go ahead and use the formula to calculate the probability of a binomial distribution this is if you are given like any of these probabilities uh uh i think only 0.9 has 0.91 up until 99 the others they are just in between uh they are in tens or fives multiple of five so it's 80 85 75 and 30 yeah so if they give you the the other ones that are in between there or there or there or there you just need to use the formula but it will be highly unlikely that they will do that to you lucky they might give it to you in the assignment but not in the exam in the exam they will prefer to use the ones on the table but in the assignment so that they test on the knowledge of using tables and also using your formulas they might give you those ones okay um let's say for instance you have a probability of 0.8 and an x value of um zero so from the table how do you read that do you go to the one that's at the bottom so what will be your n let's start there what will be your n your n will be six your n will be six so if you go to n six sorry my bad let's make it smaller so it will be on the first page where n is six so i need to rotate the table just hold on hold on to that so our n our n is six and you said the probability is equals to 0.8 so which means it's 0.80 so it means it will be at the bottom of the page so we will go to 0.20 at the top because we know that 0.8 and 0.20 are complement of one another so it will be on on this 0.80 so and you want the probability of x is equals to 0 right yes since it's at the bottom here you're going to go to the right and say any six x is zero and that probability will be 0.001 okay no no now i get it yes you don't use this site because if you use this site you will see that the probability will be wrong it will be 0.2621 so you be careful my mistake was i was going directly to the table where it says 0.80 that one at the bottom and i started reading from this so now i see you only have to rotate the one that's at the top but remember this one this one also now is not rotated you can see that it starts from the right will be the site it starts from 7 8 9 10 0.8 will be there but the value that i'm looking for for the n is not here the same way on this one is 20 so this table also has some 11 12 18 14 there are other probability tables so on on this they cut them off and then they only give you those three pages but there are other pages as well there is probability of 15 or where n is 15 n is 16 n is 17 but they don't exist on this now um so you just need to be very careful when you use the table because this part and that part are one thing they are one table combined it's just that they are printed on two separate pages so they cannot actually get it yeah okay if there are no other questions i'm going to post the video